A356144 Coefficients of the set of partition polynomials [RT] = [P][E]; i.e., coefficients of polynomials resulting from using the set of refined Eulerian polynomials, [E], of A145271 as the indeterminates of the set of permutahedra polynomials, [P], of A133314. Irregular triangle read by rows with lengths given by A000041.

1, -1, 1, -1, -1, 2, -1, 1, -3, 2, 1, -1, -1, 4, -4, -2, 5, -1, -1, 1, -5, 8, 2, -4, -2, -4, 5, 4, -4, -1, -1, 6, -12, -3, 8, 18, -6, -14, 13, 2, -16, 14, 0, -8, -1, 1, -7, 18, 3, -20, 0, -15, 8, 18, 57, 6, -54, -15, -12, 84, -30, -48, 14, 14, -8, -13, -1, -1, 8, -24, -4, 32, 51, -27, -16, -6, 171, -42, -177, 50, 90, -18, 456, -276, -246, -15, 30, 154, -42, 124, -166, -113, 42, 6, -21, -19, -1

Offset: 0

Author

Tom Copeland, Jul 27 2022

Keywords

• sign
• tabf

Comments

I stipulate that the row lengths are A000041, but this imposes the insertion of a zero as a coefficient of a monomial for the polynomial RT_7 and for RT_8. The number of nonzero coefficients in each higher order polynomial remains to be determined. The monomials of the partition polynomials are arranged in the order (bottom to top) in Abramowitz and Stegun (starting on p. 831, link in A000041).

The analytic interpretation of these coefficients is related to the e.g.f.s of reciprocals of the derivatives (slopes of tangents) of a pair of compositionally inverse e.g.f.s as explicitly shown in the formulas.

With the notation introduced in the formula section, this set of partition polynomials, [RT], is the e.g.f. counterpart to the special Schur expansion coefficients [b], or [K], of A355201 for o.g.f.s. and is conjugate dual to the Lagrange inversion polynomials [L] of A134685.

For example, as shown in the formulas, [RT] = [P][E] = [P][L][P] where [P] is the set of polynomials of A133314, the refined Euler characteristic polynomials of the permutahedra; [E], the set A145271, the refined Eulerian polynomials; and [L], the set A134685, the classic Lagrange inversion polynomials--all related to transformations of e.g.f.s, or Taylor series, for which [RT], [L], and [E] can each be used to give the compositional inverse and [P], the multiplicative inverse, or reciprocal.

On the other hand, as shown in formulas for A355201, [K] = [R][N] = [R][A][R] where [R] is the set A263633 (mod signs), refined Pascal polynomials; [N], the set A134264, the refined Narayana, or noncrossing partition, polynomials; and [A], the set A133437, the refined Euler characteristic polynomials of the associahedra--all related to transformations of o.g.f.s, or power series, for which [K], [A], and [N] can each be used to give the compositional inverse and [R], the multiplicative inverse, or reciprocal. This is related to three pairs of compositionally inverse series--two pairs of Laurent series and one pair of power series.

Examples

			Arranged by rows, the coefficients are
0)  1;
1) -1;
2)  1, -1;
3) -1, 2, -1;
4)  1, -3, 2, 1, -1;
5) -1, 4, -4, -2, 5, -1, -1;
6)  1, -5, 8, 2, -4, -2, -4, 5, 4, -4, -1;
7) -1, 6, -12, -3, 8, 18, -6, -14, 13, 2, -16, 14, 0, -8, -1;
8)  1, -7, 18, 3, -20, 0, -15, 8, 18, 57, 6, -54, -15, -12, 84, -30, -48, 14, 14, -8, -13, -1;
. . .
The first few partition polynomials are
RT_0 =  1,
RT_1 = -a1,
RT_2 = a1^2  - a2,
RT_3 = -a1^3 + 2 a1 a2 - a3,
Rt_4 = a1^4 - 3 a1^2 a2 + 2 a2^2 + a1 a3 - a4,
RT_5 = -a1^5 + 4 a1^3 a2 - 4 a1 a2^2 - 2 a1^2 a3 + 5 a2 a3 - a1 a4 - a5,
RT_6 = a1^6 - 5 a1^4 a2 + 8 a1^2 a2^2 + 2 a1^3 a3 - 4 a2^3 - 2 a1 a2 a3 - 4 a1^2 a4 + 5 a3^2 + 4 a2 a4 - 4 a1 a5 - a6,
RT_7 = -a1^7 + 6 a1^5 a2 - 12*a1^3 a2^2 - 3 a1^4 a3 + 8 a1 a2^3 + 18 a1^2 a2 a3 - 6 a1^3 a4 - 14 a2^2 a3 + 13 a1 a3^2 + 2 a1 a2 a4 - 16 a1^2 a5 + 14 a3 a4 + 0 a2 a5 - 8 a1 a6 - a7,
RT_8 =  a1^8 - 7 a1^6 a2 + 18 a1^4 a2^2 + 3 a1^5 a3 - 20 a1^2 a2^3 + 0 a1^3 a2 a3 - 15 a1^4 a4 + 8 a2^4 + 18 a1 a2^2 a3 + 57 a1^2 a3^2 + 6 a1^2 a2 a4 - 54 a1^3 a5 - 15 a2 a3^2 - 12 a2^2 a4 + 84 a1 a3 a4 - 30 a1 a2 a5 - 48 a1^2 a6 + 14 a4^2 + 14 a3 a5 - 8 a2 a6 - 13 a1 a7 - a8.
		

Links

• Tom Copeland, One Matrix to Rule Them All, 2022.

Crossrefs

Cf. A000108, A133314, A133437, A134685, A145271, A263633, A355201, A356145.

Programs

Mathematica

rows[nn_] := {{1}}~Join~With[{s = 1 / D[InverseSeries[Integrate[1/(1 + Sum[c[k] x^k/k!, {k, nn}] + O[x]^(nn+1)), x]], x]}, Table[Coefficient[n! s, x^n Product[c[t], {t, p}]], {n, nn}, {p, Reverse[Sort[Sort /@ IntegerPartitions[n]]]}]];
rows[7] // Flatten (* Andrey Zabolotskiy, Feb 17 2024 *)

SageMath

B. = PolynomialRing(ZZ)
A. = PowerSeriesRing(B)
f = 1/(1 + a1*x + a2*x^2/factorial(2) + a3*x^3/factorial(3) + a4*x^4/factorial(4) + a5*x^5/factorial(5) + a6*x^6/factorial(6) + a7*x^7/factorial(7) + a8*x^8/factorial(8) + a9*x^9/factorial(9) + a10*x^10/factorial(10) )
g = integrate(f)
h = g.reverse()
w = derivative(h,x)
I = 1 / w
# Added by # Peter Luschny, Feb 17 2024:
# The list of coefficients in sparse format (i.e. without the zeros):
for n, c in enumerate(I.list()[:10]):
    print(f"RT[{n}]", (factorial(n)*c).coefficients())

Formula

Denote this set of partition polynomials by [RT], the permutahedra polynomials of A133314 by [P], the refined Eulerian polynomials of A145271 by [E], and the Lagrange inversion polynomials of A134685 for e.g.f.s by [L]. Let the typically noncommutative product of two sets, e.g., [P][E], represent the substitution of the polynomials of [E] for the indeterminates of [P], i.e., a composition at the level of the indeterminates (see A356145 for examples). Let [I] be the substitutional identity transformation, and mark the substitutional inverse with the superscript -1. Then the following relations hold.

[RT] = [P][E] = [P][L][P] = [P]^{-1}[L][P] = [P][L][P]^{-1} since [P] is an involution, i.e., [P]^2 = [I], or [P] = [P]^{-1}, so [RT] and [L] are conjugate duals.

[RT]^{-1} = ([P][E])^{-1} = [E]^{-1}[P] = ([P][L][P])^{-1} = [P][L][P] = [RT], with [E]^{-1} = A356145, since [L] and [P] are involutions, so is [RT], i.e., [RT]^2 = [I].

RT_n(a_1,a_2,...,a_n) = D_{x=0}^n 1 / [ D_x f^{(-1)}(x)] for which D_x is the derivative w.r.t. x and the indeterminates are defined by 1 / [D_x f(x)] = 1 + a_1 x + a_2 x^2/2! + a_3 x^3/3! + ... with f(x) and f^{(-1)}(x) a compositional inverse pair of formal Taylor series, or e.g.f.s. This is the analytic equivalent of the algebraic relation [RT] = [P][E]. In words, the partition polynomials of row n (initial row is 0) is the n-th coefficient of the formal Taylor series of the reciprocal of the derivative of the compositional inverse of a function in terms of the Taylor series coefficients of the reciprocal of the derivative of that function. Note the correspondence with the analytic interpretation of [E]^{-1} of A356145, consistent with the algebraic identities above.

RT_n(a_1,a_2,...,a_n) = D_{x=0}^n f'(f^{(-1)}(x)) also, by the inverse function theorem, where the prime denotes differentiation with respect to the argument of the function.

With all a_k = (-1)^k, RT_0 = RT_1 = 1, otherwise RT_n = 0. This is determined with f(x) = e^{x}-1 and f^{(-1)}(x) = log(1+x).

With all a_k = 1, RT_0 = 1, RT_1 = -1, otherwise RT_n=0. This is determined with f(x) =1-e^{-x} and f^{(-1)}(x) = -log(1-x).

With all a_k = -1, RT_0 = 1 and RT_n = 2^(n-1) otherwise. This is determined with f(x) = (x - log(2-e^x))/2 and f^{(-1)}(x) = x - log(cosh(x)). (Careful, these are not the row sums of the absolute values of the numerical coefficients, which for the first ten polynomials are 1, 1, 2, 4, 8, 18, 40, 122, 446, and 2428.)

With a_k = k! 2^k, RT_0 = 1 and RT_n = -2*(2(n-1))! / (n-1)! = -2*n!*A000108(n-1) otherwise. This is determined with f(x) = x - x^2 and f^{(-1)}(x) = (1 - sqrt(1-4x))/2. Similar relations hold for the Fuss-Catalan sequences with f(x) = x - x^{m+1} for m > 1.

Extensions

Order of terms in rows 4-6 corrected by Andrey Zabolotskiy, Feb 17 2024